The Small Angle X-ray Scattering Technique: An...

Dr. Gianluca Croce 1

The Small Angle X-ray Scattering Technique:An Overview

Dr. Gianluca Croce, Ph.DDISTA - Univ. Piemonte OrientaleVia T. Michel 11,15121 Alessandria (Italy)[email protected]


X-ray scattering techniques are a family of non-destructive analytical techniqueswhich reveal information about the crystallographic structure, chemicalcomposition, and physical properties of materials and thin films. These techniquesare based on observing the scattered intensity of an X-ray beam hitting a sampleas a function of incident and scattered angle, polarization, and wavelength orenergy.

Materials that do not have long range order may also be studied by scatteringmethods that rely on elastic scattering of monochromatic X-rays:

Small angle X-ray scattering (SAXS) probes structure in the nanometer tomicrometer range by measuring scattering intensity at scattering angles 2θclose to 0°

An Overview

Small-angle X-ray scattering (SAXS) is a small-angle scattering (SAS) techniquewhere the elastic scattering of X-rays (wavelength 0.1 ... 0.2 nm) by a samplewhich has inhomogeneities in the nm-range, is recorded at very low angles(typically 0.1 - 10°). This angular range contains information about the shape andsize of macromolecules, characteristic distances of partially ordered materials,pore sizes, and other data. SAXS is capable of delivering structural information ofmacromolecules between 5 and 25 nm, of repeat distances in partially orderedsystems of up to 150 nm.

(Glatter, O.; O. Kratky (1982). Small Angle X-ray Scattering. Academic Press.http://physchem.kfunigraz.ac.at/sm/Software.htm)

An Overview


Until the 1970s, SAXS experiments were done on instruments equipped bylaboratory X-ray tubes. Nowadays the X-ray source can be also a synchrotronlight which provides a higher X-ray flux. Also neutrons can be employed.

Conceptually, a SAXS experiment is simple: a sample is illuminated by X-raysand the scattered radiation is registered by a detector.

A SAXS Instrument


A SAXS Instrument

The major problem that must be overcome in SAXS instrumentation is theseparation of the weak scattered intensity from the strong main beam.


Laboratory SAXS instruments can be divided into two main groups:

1. Point-collimation instruments have pinholes that shape the X-ray beam to a smallcircular or elliptical spot that illuminates the sample. The scattered intensity is smalland therefore the measurement time is in the order of hours or days in case of veryweak scatterers.

2. Line-collimation instruments confine the beam only in one dimension so that thebeam profile is a long but narrow line. The illuminated sample volume is much largercompared to point-collimation and the scattered intensity at the same flux density isproportionally larger. Thus measuring times with line-collimation SAXS instrumentsare much shorter compared to point-collimation and are in the range of minutes tohours.

A SAXS Instrumentation

The camera length is variable between 1 and 10 m

Synchrotron SAXS: Data can be collected in the minute/(milli)second time regime.


SAXS Beamline

SAXIER: Small-Angle X-Ray Scattering Initiative for Europe

• The X33 beamline for biological small-angle scattering. EMBL/DESY, Hamburg,Germany

• The ID13 beamline ESRF, Grenoble, France

• The small- and wide-angle scattering beamline SWING. SOLEIL, France

• The Austrian small-angle scattering beamline BL 5.2 L. ELETTRA, Italy

• The non-crystalline diffraction beamline 2.1 in long geometry.SRS Daresbury, UK

http://www.saxier.org/


The detector and data acquisition system allows measurements in one- andtwo-dimensional manner.


1D 2D


The scattering pattern contains theinformation on the structure of the sample.

??????


Why Small Angle?

3) Any scattering process is characterized by a reciprocity law which givesan inverse relationship between object size and scattering angle

1) When an X-ray passes through an object it causes electrons to oscillateto the same frequency as that of the incident wave

2) These oscillating charges give rise coherent secondary electromagneticwaves with the same wavelenght as that of the incident beam


1 2

Colloidal dimensions (between tens and several thousand Å) are enormouslylarge compared to the X-ray wavelength (i.e. CuKa =1.54A)

Why Small Angle?

!

object"1

angle


110µm11µm400

……….

68nm6.8nm0.23

44nm4.4nm0.15

0.1°1°

Probed dimensionλ[nm]

Why Small Angle?

The SAXS technique is a tool which permits us to measure thescattered intensity at angle smaller than one degree

Diffraction < SAXS < Optical Microscope


SAXS vs. TEM Trasmission Electron Microscopy (TEM) cover the same dimensionalrange investigated by SAXS

The two techniques are complementary.

WHY?????



The two techniques are complementary

TEM good for:

Direct and detailed image

Local details

Local surface

Faithfully represents local complexities

TEM not good for:

Tiny image

Statistically significant average informationSAXS



The two techniques are complementary

SAXS good for:

Global parameters and distribution

Different sample states(solid, wet orsolutions samples…)

No destructive/artifacts in samplepreparation

In situ transition study


SAXS vs. TEM Transmission Electron Microscopy (TEM) cover the same dimensionalrange investigated by SAXS

The two techniques are complementary:

- TEM needs SAXS to obtain significant sampling and makequantitative statements

- SAXS needs TEM for clues about the shape of particles for use indata analysis


What is an object? What SAXS ‘sees’ as objects are the spatial variations in the electrondensity inside the irradiated portion of matter

Electronic density difference is also called contrast

An example: A homogenous particle (i.e. a macromolecule) with aconstant electron density dispersed in a matrix (i.e. a solvent) with a

different electron density

particle

matrix


What is an object? On an atomic scale, the electron density continuously varies from highervalues in the area surrounding an atomic position to lower values in thespace between atoms


What is an object? On an atomic scale, the electron density continuously varies from highervalues in the area surrounding an atomic position to lower values in thespace between atoms

The SAXS explores dimensions of few nanometers. This implies thatSAXS is not sensitive to electron density fluctuation on an atomic scale.

Consequently a portion of matter of homogenous composition isconsidered as a homogenous particle (a particle with constant electrondensity)

particle

matrix


Adman, E. T., Godden, J. W., Turley,J Biol Chem 270 pp. 27458 (1995)

Hao Q, et al., Acta CrystallographicaD55, (1), 243-246 (1999)


What is an object? The scattering is a good probe for matter with higher electron densitycontrast

The SAXS technique is good for:

- amorphous particle

- crystalline particle

- homogeneous material with even voids(pores, blisters, cracks and crazes)

particle

matrix

!


•SAXS is used for the determination of the microscale or nanoscale structure ofparticle systems in terms of such parameters as averaged particle sizes, shapes,distribution, and surface-to-volume ratio.

•The materials can be solid or liquid and they can contain solid, liquid or gaseousdomains (so-called particles) of the same or another material in any combination.

•Not only particles, but also the structure of ordered systems like lamellae, andfractal-like materials can be studied.

•The method is accurate, non-destructive and usually requires only a minimum ofsample preparation.

•Applications are very broad and include colloids of all types, metals, cement, oil,polymers, plastics, proteins, foods and pharmaceuticals and can be found inresearch as well as in quality control.


LIQUIDProtein; Pharmaceuticals.

SOLIDPowder; Liquid Crystal;

Nano Materials; Polymer; Fiber.


Single Particle Scattering The scattering of a single particle in a matrix is simply its Fourier transform.

FT

FT


Single Particle Scattering The Fourier trasform can be calculated analytically for a large number ofshape:

a b Ellipsoidal particle with constant orientation

!

I0(h,a,b) = "#24

3$ab2

%

& '

(

) *

23sin x + x cos x

x3

%

& '

(

) *

2

The scattering intensity I0(h):

Where:

!

x = a2h1

2+ b

2h2

2

h h =4" sin#

$

%

& '

(

) * Scattering vector with h1 and h2 component on the dector plane


Single Particle Scattering

a b

!

I0(h,a,b) = "#24

3$ab2

%

& '

(

) *

23sin x + x cos x

x3

%

& '

(

) *

2

Electron Density Difference

(particle/matrix)

Volume of the ellipsoidBivariate function (surface) whosecontour lines have an ellipsoidal shapeoriented perpendicularly to the particle


ab

h1

h2

Bivariate function (surface) whose contour lineshave an ellipsoidal shape oriented

perpendicularly to the particle

!

x = a2h1

2+ b

2h2

2

h h =4" sin#

$

%

& '

(

) *

Scattering vector with h1 and h2component on the detector plane

!

3sin x " x cos x

x3

#

$ %

&

' (

2



ab

h1h2


This last term is a consequence of the reciprocal law between thescattering particle and its scattering pattern

OR

Between a function and its Fourier transform


!

object"1

angle

Remember:

•Line sections of the scattering patterns (=linear detector) of an ellipsoid ofdifferent shapes

•The curves scattering decrease inversely to the vector h1



Log(I0)

Single Particle Scattering •A 1D scattering curve theoretically is composed by a central peak with aseries of rapidly damped peaks of much smaller intensity

•For simplicity is useful to represent the data in a logarithmic scale tobetter appreciate the small intensity peaks


Single Particle Scattering •Scattering equation of particles of different shape:


Single Particle Scattering •1D scattering curve of particles of different shape and equal dimensions


Group of Particle Scattering In a real case we normally study system composed by many particlesdispersed in a matrix (i.e. a solvent)

A dilute system of ellipsoidal particle with the same orientation:

• Same dimension: a simple sum of the scattering of each single particle

• Different dimension: multiply the function for the scattering of a singleparticle by the distribution of sizes


Group of Particle Scattering What does it mean “Multiply the function for the scattering of a single particleby the distribution of sizes”?

!

I0(h,a,b) = D(a,b)I

0(h,a,b)dadb""

!

I0(h,a,b) = "#24

3$ab2

%

& '

(

) *

23sin x + x cos x

x3

%

& '

(

) *

2

Function of the semi-axes of the ellipsoid; a and bparameters control the dimensions of the particles

Equation with described the scattering for a single ellipsoid:


Group of Particle Scattering If the orientation of the particles vary or if their orientation changes over time(random orientation):

The scattering is averaged in all directions and the pattern will be isotropic


Group of Particle Scattering In this ‘real’ situation, a distribution of directions should be introduced formimic the scattering pattern I(h,a,b)

Obviously the information on particle shapes and dimensions is mixed and it’snecessary to make assumptions about their shape or size distribution


Group of Particle Scattering


When particles are very close one to another

Group of Particle Scattering(Densely Packed)

They cause a constructive inference similar to the Bragg peaks(diffraction).

If these distance have somedegree of regularity


•A SAXS signal contain information about shape and size particles (alsodistances)

•These information are confused in a SAXS pattern

•These contributions are impossible to separate without externalinformation (i.e. particle shape, a certain distribution, etc…)

Data Analysis

Data Analysis Manipulation

•Retrieve particles information processing experimentaldata by means of mathematical transformations to aassumed shape

•From an assumed particle distribution, calculate thescattering intensity in order to try to fit the experimental data


At smallest angle, the scattering curve of a dilute system (similar sizeparticles) can be approximated:

Guinier Approximation

!

I(h)"e#aR

g

2h2

Gaussian Function

Where:Rg is called Gyration Radiusa is a factor depending on particle orientation(i.e. random orientation a=1/3)


The Gyration Radius is defined in analogy to the radius of inertia in mechanics.

R is defined as the mean square distance from the centre of gravity (R=√r2) wherethe role of the ‘mass’ is played by the electrons.


!

I(h) = I(0)e

"R2h2

3

The equation of randomly oriented particle (a=1/3) is:

lnI(h

)h2

Rg can be calculated from theslope of the strain region of the plot




• The Rg is a parameter doesn’t provide any geometrical information of particles.

• It’s always necessary to consider an ‘a priori’ geometry (shape) of the particles.

• If the particle shape is defined, the Guinier approximation provides a good indicationof average size particles.


The Guinier approximation works well when, in a dilute system, the particles preservethe same shape with also different dimension (in a limited range)

If the particle dimensions are drastically different may occur to experimentally observetwo definite Guinier region (determinable separately)

Guinier RegionLnI(h)

h^2


• Examining the queues of a SAXS signal, Porod observed that in most case the datafollow an asymptotic behavior with a fourth powerlaw decay

• This behavior is expected for so-called two-phase system (when the el. density ρ canassume only two value: ρ1 and zero)

Porod Law


Mathematically for large value of h:

Porod Law

2

114)(

2)( !!

"#= S

hhI

Porod Plot

h^4I(h)

h

In an ideal case (spherical particle with ρ=ρ1;0) the Porod Plot trends to a plateau.This constant value is proportional to the surface area at the interface.

Where S represents the surface area of the particles andmay be determined from experimental data.


Porod also demonstrated that normalizing the intensities (Absolute Intensity), thefollowing relation for a two-phase system is possible:

Integral Quantities

!

1

Vh2I(h)dh = v(1" v)(#

2"

0

$

% #1)2

V = scattering volume

ρ1;ρ2 = electronic density of two phases

v;1-v = volumetric fraction of two phase

Scattering Power

This quantity is calculable from the experimental data if it’s possible toextrapolate them to zero (Guinier) and to infinity (Porod)

This equation permit to estimate the volumetric fraction knowing the electronic density,and viceversa, of a two-phase system.


• Glatter, O. & Kratky, O., eds. Small Angle X-ray Scattering.Academic Press, 1982. ISBN 0-12-286280-5Free available on-line (Short link: http://tinyurl.com/ktukhf)

• L.A. Feigin & D.I. Svergun: Structure Analysis by Small-Angle X-Ray and NeutronScattering.New York: Plenum Press, 1987. ISBN 0-306-42629-3Free available on-line (Short link: http://tinyurl.com/lwcfwj)

• Lipfert J, Doniach S. Small-angle X-ray scattering from RNA, proteins, and proteincomplexes. Annu Rev Biophys Biomol Struct. 2007;36:307-27.

•SAXIER: Small-Angle X-Ray Scattering Initiative for Europe (http://www.saxier.org/)

Bibliography

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